While the number of constraints may be extremely large, each one is a standard linear inequality constraint, which implies that the feasible region is convex. Since we have a constraint oracle that runs in polynomial time,Finding the two shortest simple paths between two nodes takes \(O(NM)\)  time using Yen's algorithm [1]}. If \(\delta =0\)  and edge weights are strictly positive, we can use an algorithm not restricted to simple paths that runs in \(O(M+N\log {N})\)  time [2]}. this system can be optimized in polynomial time regardless of the number of constraints. Using the ellipsoid algorithm introduced by Khachiyan (see [3]}), we can solve a linear program within finite precision in a polynomial number of iterations [4]}. This results in the following proposition.
